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ELEC97092 (EE4-45) Wavelets, Representation Learning and their Applications

Lecturer(s): Prof Pier-Luigi Dragotti


Finding useful information in huge amount of data is as difficult as finding a needle in a haystack. The key insight of wavelet theory is that by finding alternative representations of signals, it is possible to extract their essential information in a fast and effective way. Wavelet theory provides the tools to find alternative representations of a signal and then to choose the representation which is more appropriate for the task at hand. Because of this flexibility, the wavelet transform
play a pivotal role in many modern data-driven applications such as time-series analysis, multimedia processing and applications in the bio-medical domain. It is also at the heart of methods to learn the representation of a signal through data.

The main aim of the course is to introduce students to wavelet theory. Students will learn about Hilbert spaces and the notions of signal approximation and projection. They will learn about orthogonal, biorthogonal and redundant representations and how to obtain some of these representations using filter banks.

Students will learn how to design perfect-reconstruction filter banks and how to relate these constructions to the multi-resolution properties inherent to wavelets. The course will also cover some applications in which wavelets have been successful like image compression, image super-resolution and in neuroscience.

Finally, we will use the principles behind wavelet theory to understand representation learning in particular dictionary learning.

Learning Outcomes

Knowledge and understanding
- understanding the fundamentals of wavelet theory and Hilbert Spaces
- familiarity with the most commonly used wavelets (e.g Daubechies wavelets)
-understanding how to design perfect reconstruction filter banks
- understanding the link between design of filter banks and construction of discrete and continuous-time bases for efficient signal analysis.
- basic knowledge of image and video compression principles
-Dictionary learning


Part I: Introduction and Background
1. Motivation: Why wavelets, subband coding and multiresolution analysis? Mathematical background.
Hilbert spaces. Unitary operators. Review of Fourier theory. Continuous and discrete time signal
2. Time-frequency analysis. Multirate signal processing. Projections and approximations.

Part II: Discrete-Time Bases and Filter Banks
3. Elementary filter banks.
Analysis and design of filter banks. Spectral Factorization. Daubechies filters.
4. Orthogonal and biorthogonal filter banks.
Tree structured filter banks. Discrete wavelet transform. Multidimensional filter banks.

Part III: Continuous-Time Bases and Wavelets
5. Iterated filter banks. The Haar and Sinc cases.
The limit of iterated filter banks.
6. Wavelets from Filters. Construction of compactly supported wavelet bases.
Regularity. Approximation properties. Localization.
7. The idea of multiresolution. Multiresolution analysis.

Part IV: Applications
8. Fundamentals of compression. Analysis and design of transform coding systems. Image Compression, the new compression standard (JPEG200) and the old standard. Why is the wavelet transform better than the discrete cosine transform? Advanced topics: Beyond JPEG2000, non-linear approximation and compression.
9. Modern sampling theory: Shannon sampling theorem revisited, sampling parametric not bandlimited signals, multichannel sampling and image super-resolution.

Part V: Advanced topics
10. Dictionary Learning
11 Invertible Neural Networks and the lifting scheme
Exam Duration: 3:00hrs
Coursework contribution: 25%

Term: Autumn

Closed or Open Book (end of year exam): Closed

Coursework Requirement:
         To be announced

Oral Exam Required (as final assessment): N/A

Prerequisite module(s): None required

Course Homepage:

Book List:
Please see Module Reading list